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Martingale methods in financial modelling. (English) Zbl 0906.60001
Applications of Mathematics. 36. Berlin: Springer. xii, 512 p. (1997).
This book is a comprehensive and up-to-date presentation of the martingale approach for pricing and hedging derivative securities. It consists of two main parts. The first half of the book deals with the classical concepts of pricing and hedging of contingent claims in financial markets. Chapter 1 briefly reviews important kinds of financial contracts. In a one-period model, the basic notions of arbitrage pricing theory, like replicating portfolios and martingale measures, are introduced. In the framework of the well-known Cox, Ross and Rubinstein (CRR) binomial model for an asset price process, Chapter 2 provides the valuation formula for European call- and put-options. The framework of Chapter 2 is the well-known CRR binomial model for an asset price process. The CRR option valuation formula for European call- and put-options are derived once by a replication argument and then by a purely probabilistic argument via the equivalent martingale measure. It is also shown that the famous Black-Scholes formula can be obtained from it by an asymptotic procedure using a properly chosen sequence of binomial tress. Then American call- and put-options are evaluated by no-arbitrage arguments. The chapter concludes with the evaluation of options in the presence of transaction costs.
Chapter 3 deals with finite security markets, i.e., general discrete-time models with finite time horizon and finite probability space. Trading strategies and arbitrage opportunities are defined, and by elementary arguments a version of the fundamental theorem of asset pricing is proved. This states that the absence of arbitrage is equivalent to the existence of an equivalent martingale measure for discounted asset prices. Furthermore, it is shown that completeness of an arbitrage-free market model is equivalent to the uniqueness of the equivalent martingale measure. Chapter 4 examines various types of market imperfections in the finite security market model, and, i.e., completeness of the market, restrictions of short-selling of stocks and borrowing of cash, and finally the case of different borrowing and lending rates. The concept of mean-variance hedging is discussed.
Chapter 5 establishes the classical Black-Scholes formula for European call- and put-options when the stock price follows a one-dimensional geometric Brownian motion. It is first derived via the partial differential equations approach and then via the equivalent martingale approach. A short sensitivity analysis of the market parameters is also included. Chapter 6 presents various modifications of the one-dimensional Black-Scholes model. The classical Black futures formula is derived, and the standard Black-Scholes formula is extended to the case of options on divided paying stocks. The last section briefly addresses problems and techniques related to the volatility of asset prices. Chapter 7 prices various foreign market derivatives by the martingale approach in the following framework: the foreign stock price and the exchange rate follow geometric Brownian motions, and the domestic and foreign risk-free interest rate are nonnegative constants. Chapter 8 addresses the pricing and hedging of American options and explains its link to the theory of optimal stopping for diffusion processes.
Chapter 9 serves as an excellent reference for pricing exotic options as numerous examples of them are analyzed in the Black-Scholes framework. The first section of Chapter 10 deals with a general continuous-time model in which asset prices are driven by continuous semimartingales. Arbitrage opportunities and ‘reasonable’ classes of admissible trading strategies are introduced in this general set-up. This section also contains a nice discussion on deep theoretical results concerning the fundamental theorem of asset pricing. The second section deals with the multi-dimensional Black-Scholes model. It focuses on the problem of market completeness and mean-variance hedging in incomplete markets.
The second half of this book is devoted to fixed-income markets and more precisely to term structure models and the pricing of interest rate derivatives. In Chapter 11, various interest rate sensitive instruments such as zero-coupon bonds, coupon-bearing bonds, interest rate futures and interest rate swaps are introduced. Chapter 12 gives a survey of the most popular short-term interest rate models. In Chapter 13, the Heath-Jarrow-Morton (HJM) approach to term structure modelling is presented. There the dynamics of instantaneous continuously compounded forward rates are exogeneously specified. The forward measure approach is then applied to evaluate contingent claims by absence of arbitrage. Chapter 14 deals with models of bond prices and LIBOR rates. After presenting general properties of arbitrage-free families of bond prices, models of forward LIBOR rate and forward swap rates are extensively discussed. In Chapter 15, the forward measure methodology is employed in arbitrage pricing of interest rate derivatives in a Gaussian framework, i.e. in models in which the bond price volatilities are deterministic functions. In a Gaussian HJM framework and in the case of lognormal models of forward LIBOR and swap rates, Chapter 16 provides explicit valuation solutions for swap derivatives In Chapter 17, the arbitrage valuation of foreign market derivatives is extensively studied.
On the whole this book provides a wide range of topics and will appeal to both practitioners and mathematicians. When only special cases or models are provided, the authors give useful references that will help researchers to obtain even more insight in the topics. Especially the second part gives an excellent introduction and survey of the state of the art in research and will clearly become a standard reference. Thus, this book can be strongly recommended to any reader learning, teaching or working in the field of mathematical finance.

##### MSC:
 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 62P05 Applications of statistics to actuarial sciences and financial mathematics 91B28 Finance etc. (MSC2000) 60Hxx Stochastic analysis